On the K-theory of pullbacks
By Markus Land, Georg Tamme. Annals of Mathematics, 2018
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To any pullback square of ring spectra we associate a new ring spectrum and use it to describe the failure of excision in algebraic $K$-theory. The construction of this new ring spectrum is categorical and hence allows to determine the failure of excision for any localizing invariant in place of $K$-theory. As immediate consequences we obtain an improved version of Suslin's excision result in $K$-theory, generalizations of results of Geisser and Hesselholt on torsion in (bi)relative $K$-groups, and a generalized version of pro-excision for $K$-theory. Furthermore, we show that any truncating invariant satisfies excision, nilinvariance, and cdh-descent. Examples of truncating invariants include the fibre of the cyclotomic trace, the fibre of the rational Goodwillie--Jones Chern character, periodic cyclic homology in characteristic zero, and homotopy $K$-theory. Various of the results we obtain have been known previously, though most of them in weaker forms and with less direct proofs.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/9d3b3b723ce285d5b6031aa51a158283352bd1f4
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- K-theory and topological cyclic homology of henselian pairs (2018, background): “nd similarly for any other functor on rings. The following was proved in the rational case in [19], with finite coefficients in [31], integrally under some assumptions in [23], and in full generality in [51]. Here commutativity is not necessary. K-THEORY AND TOPOLOGICAL CYCLIC HOMOLOGY OF HENSELIAN PAI”